1 Dependencies and settings

# R packages
library("corrr") # 0.4.2
library("cowplot") # 1.0.0
library("dplyr") # 1.0.0
library("DT") #0.14
library("ggplot2") # 3.3.1
library("oem") # 2.0.10
library("gridExtra") # 2.3
library("knitr") # 1.28
library("kableExtra") # 1.1.0
library("magrittr") # 1.5
library("mice") # 3.9.0
library("pROC") # 1.16.2
library("purrr") # 0.3.4
library("rcompanion") # 2.3.25
library("rms") # 6.0.1
library("splines") # base
library("tableone") # 12.0.0
library("tibble") # 3.0.1
library("tidyr") # 1.1.0

# Namespace clash
select <- dplyr::select

# Settings
set.seed(123)
theme_set(theme_bw())
n_imputations <- 2    # Number of multiple imputations with MICE
n_boot_val <- 40      # Number of bootstraps with rms validation and calibration
n_boot_probs <- 100   # Number of bootstraps for computing predicted probabilitiy CIs
n_rep <- 2            # Number of repeats with grouped lasso
CACHE <- TRUE         # Whether to use cached results for long computations

2 Data

We begin by loading the “training” dataset and restrict to (i) patients age 18 and older and (ii) with an index date more than 2 weeks prior to the data release. Note that we are not unfortunately allowed to publicly share this data.

2.1 Load

train_data <- readRDS("train_data.rds")  %>%
  filter(age >= 18) %>%
  filter(as.Date("2020-06-05") - index_date > 14)

2.2 Clean

# Recoding
train_data <- train_data %>% 
  mutate(
    ## "died" in add_deaths() should be an integer
    died = as.integer(died),
    
    ## Convert unknown or other/unknown to missing
    race = ifelse(race == "Other/Unknown", NA, race),
    ethnicity = ifelse(ethnicity == "Unknown", NA, ethnicity),
    sex = ifelse(sex == "Unknown", NA, sex),
    
    ## Oxygen saturation should have plausible values
    spo2 = ifelse(spo2 == 0, NA_real_, spo2),
    spo2 = ifelse(spo2 > 100, 100, spo2)
  ) %>%
  
  ## Add option for comorbidities in create_comorbidities() to be character
  rename(diabunc = diab) %>%
  mutate_at(
    c("ami", "chf", "pvd", "cevd", "dementia", "copd", "rheumd", "pud",
      "mld", "diabunc", "diabwc", "hp", "rend", "canc", "msld", "metacanc",
      "aids", "hypc", "hypunc"),
    function (x) ifelse(x == 1, "Yes", "No")
  ) 

# Create "derived" variables

## Function to add race/ethnicity variable to dataset (done after separately 
## imputing missing race + ethnicity information)
add_race_ethnicity <- function(data){
  data %>% mutate(
    race_ethnicity = case_when(
      race == "Caucasian" & ethnicity != "Hispanic" ~ "Non-Hispanic white",
      race == "African American" & ethnicity != "Hispanic" ~ "Non-Hispanic black",
      race == "Asian" & ethnicity != "Hispanic" ~ "Asian",
      is.na(race) | is.na(ethnicity) ~ NA_character_,
      TRUE ~ "Hispanic"
    ),
    race_ethnicity = relevel(factor(race_ethnicity), ref = "Non-Hispanic white")
  )
}

train_data <- train_data %>%
  mutate(
    calendar_time = as.numeric(index_date - min(index_date)),
    index_month = as.integer(format(index_date, "%m")),
    death_month = as.integer(format(date_of_death,"%m")),
    os_days = pmax(0, date_of_death - index_date),
    
    ## Categorize BMI
    bmi_cat = case_when(
      bmi < 18.5 ~ "Underweight",
      bmi >= 18.5 & bmi < 25 ~ "Normal",
      bmi >= 25 & bmi < 30 ~ "Overweight",
      bmi >= 30 ~ "Obese",
      TRUE ~ NA_character_
    ),
    
    ## Combine comorbidities
    diab = case_when(
      diabunc == "Yes" | diabwc == "Yes" ~ "Yes",
      TRUE ~ "No"
    ),
    hyp = case_when(
      hypc == "Yes" | hypunc == "Yes" ~ "Yes",
      TRUE ~ "No"
    )
  ) %>%
  
  ## Create race/ethnicity variable
  ## Later to be created after imputation from imputed race + ethnicity
  add_race_ethnicity()

2.3 Label

# Labels
## Categorical variables
demographic_cat_vars <- tribble(
  ~var, ~varlab,
  "sex", "Sex",
  "race", "Race",
  "ethnicity", "Ethnicity",
  "division", "Geographic division",
  "smoke", "Smoking",
  "race_ethnicity", 'Race/Ethnicity'
) %>% 
  mutate(group = "Demographics")

comorbidity_vars <- tribble(
  ~var, ~varlab,
  "ami", "Acute myocardial infarction",
  "chf", "Congestive heart failure",
  "pvd", "Peripheral vascular disease",
  "cevd", "Cerebrovascular disease",
  "dementia", "Dementia",
  "copd", "Chronic pulmonary disease",
  "rheumd", "Rheumatoid disease",
  "pud", "Peptic ulcer disease",
  "mld", "Mild liver disease",
  "diabunc", "Diabetes (no complications)",
  "diabwc", "Diabetes (complications)",
  "hp", "Hemiplegia or paraplegia",
  "rend", "Renal disease",
  "canc", "Cancer",
  "msld", "Moderate/severe liver disease",
  "metacanc", "Metastatic cancer",
  "aids", "AIDS/HIV",
  "hypunc", "Hypertension (no complications)",
  "hypc", "Hypertension (complications)",
  "diab", "Diabetes", # Combines diabunc and diabwc
  "hyp", "Hypertension", # Combine hypunc and hypc
) %>%
  mutate(group = "Comorbidities")

vital_cat_vars <- tribble(
  ~var, ~varlab,
  "bmi_cat", "Body Mass Index (BMI)",
) %>%
  mutate(group = "Vitals")

cat_vars <- bind_rows(demographic_cat_vars,
                      comorbidity_vars,
                      vital_cat_vars)

## Continuous variables
demographic_continuous_vars <- tribble(
  ~var, ~varlab,
  "age", "Age",
  "calendar_time", "Calendar time"
) %>%
  mutate(group = "Demographics")

vital_continuous_vars <- tribble(
  ~var, ~varlab,
  "bmi", "Body Mass Index (BMI)",
  "dbp", "Diastolic blood pressure",
  "sbp", "Systolic blood pressure",
  "hr", "Heart rate",
  "resp", "Respiration rate",
  "temp", "Temperature",
) %>%
  mutate(group = "Vitals")

lab_vars <- tribble(
  ~var, ~varlab,
  "alt", "Alanine aminotransferase (ALT)",
  "ast", "Aspartate aminotransferase (AST)",
  "crp", "C-reactive protein (CRP)",
  "creatinine", "Creatinine",
  "ferritin", "Ferritin",
  "d_dimer", "Fibrin D-Dimer",
  "ldh", "Lactate dehydrogenase (LDH)",
  "lymphocyte", "Lymphocyte count",
  "neutrophil", "Neutrophil count",
  "spo2", "Oxygen saturation",
  "pct", "Procalcitonin",
  "tni", "Troponin I",
  "plt", "Platelet count (PLT)",
  "wbc", "White blood cell count (WBC)"
) %>%
  mutate(group = "Labs")

continuous_vars <- bind_rows(
  demographic_continuous_vars,
  vital_continuous_vars,
  lab_vars
)

# All variables
vars <- bind_rows(cat_vars,
                  continuous_vars)

get_var_labs <- function(v){
  vars$varlab[match(v, vars$var)]
}

3 Preliminary data analysis

Before starting modeling, it’s a good idea to carefully inspect the data.

3.1 Sample size

train_data %>%
  count(name = "Sample size") %>% 
  mutate(Data = "Optum training set") %>%
  select(Data, `Sample size`) %>%
  kable() %>%
  kable_styling()
Data Sample size
Optum training set 13666

3.2 Missing data

missing_df <- train_data %>%
  select(one_of(vars$var)) %>%
  mutate_all(function (x) ifelse(is.na(x), 1, 0)) %>%
  mutate(id = factor(1:n())) %>%
  pivot_longer(cols = vars$var, names_to = "var", values_to = "missing") %>%
  left_join(vars, by = "var")

3.2.1 Proportion missing

# Compute proportion missing
prop_missing <- missing_df %>%
  group_by(varlab) %>%
  summarise(prop = mean(missing))
## `summarise()` ungrouping output (override with `.groups` argument)
# Plot
ggplot(prop_missing, aes(x = varlab, y = prop)) +
  geom_bar(stat = "identity") + 
  geom_text(aes(label = formatC(prop, format = "f", digits = 2)),
          nudge_y = .03, size = 3) +
ylim(c(0, 1)) +
xlab("") +
ylab("Proportion") +
coord_flip() +
scale_x_discrete(limits = rev(sort(vars$varlab)))

3.2.2 Missing by patient

ggplot(missing_df,
       aes(x = id, y = varlab,  fill = factor(missing))) +
  geom_raster() + 
  theme(axis.title.x = element_blank(),
        axis.text.x = element_blank(),
        axis.ticks.x = element_blank(),
        axis.title.y = element_blank(),
        legend.position = "bottom") +
  scale_fill_manual(name = "Missing",
                    values = c("lightgrey", "steelblue"),
                    labels = c("No", "Yes")) +
  scale_y_discrete(limits = rev(sort(vars$varlab)))
## Warning: Raster pixels are placed at uneven vertical intervals and will be shifted. Consider using geom_tile() instead.

missing_df %>% 
  # Count of missing by patient
  group_by(id) %>%
  summarise(n_missing = sum(missing),
            prop_missing = n_missing/n()) %>%
  
  # Plot
 ggplot(aes(x = prop_missing)) +
  geom_histogram(binwidth = .03, color = "white") +
  scale_x_continuous(breaks = seq(0, 1, .05)) +
  scale_y_continuous(n.breaks = 20) +
  xlab("Proportion of covariates that are missing") +
  ylab("Count") 
## `summarise()` ungrouping output (override with `.groups` argument)

3.3 Distributions

3.3.1 Covariates

3.3.1.1 Categorical

cat_var_df <-  train_data %>%
  select(one_of("ptid", cat_vars$var)) %>%
  pivot_longer(cols = cat_vars$var, names_to = "var", values_to = "value") %>%
  left_join(cat_vars, by = "var") %>%
  filter(!is.na(value)) %>%
  group_by(var, varlab, value) %>%
  summarise(n = n()) %>%
  group_by(varlab) %>%
  mutate(freq = n / sum(n)) %>%
  ungroup() %>%
  mutate(
    nudge_x = case_when(
      freq < 0.5 ~ 0.15,
      TRUE ~ -0.15
    )
  )
## `summarise()` regrouping output by 'var', 'varlab' (override with `.groups` argument)
ggplot(cat_var_df,
       aes(x = freq, y = value)) +
  geom_point() + 
  geom_text(aes(label = formatC(freq, format = "f", digits = 2)),
            nudge_x = cat_var_df$nudge_x, size = 3.5) +
  facet_wrap(~varlab, scales = "free_y", ncol = 4) +
  xlim(0, 1) +
  xlab("Proportion") +
  ylab("") +
  theme(axis.text.x = element_text(size = 10),
        axis.text.y = element_text(size = 10),
        strip.text.x = element_text(size = 7))

3.3.1.2 Continuous

3.3.1.2.1 Box plot
pivot_continuous_longer <- function(data, vars){
  col_names <- vars$var
  train_data %>%
    select(one_of("ptid", col_names)) %>%
    pivot_longer(cols = col_names, 
                 names_to = "var", 
                 values_to = "value") %>%
    left_join(vars, by = "var") %>%
    filter(!is.na(value)) 
}
continuous_var_df <- pivot_continuous_longer(train_data, 
                                             vars = continuous_vars)

plot_box <- function(data){
  ggplot(data, 
       aes(x = varlab, y = value)) +
  geom_boxplot(outlier.size = 1) + 
  facet_wrap(~varlab, scales = "free") +
  xlab("") +
  ylab("Value") +
   theme(axis.title.x=element_blank(),
        axis.text.x=element_blank(),
        axis.ticks.x=element_blank(),
        strip.text = element_text(size = 7))
}
plot_box(continuous_var_df)

3.3.1.2.2 Histogram
plot_hist <- function(data){
  ggplot(data,
       aes(x = value)) +
  geom_histogram(bins = 40, color = "white") +
  facet_wrap(~varlab, scales = "free", ncol = 4) + 
  xlab("") + ylab("Frequency") +
  theme(strip.text = element_text(size = 7))
}
plot_hist(continuous_var_df)

3.3.1.2.3 Outliers

Visual inspection of the box plots and histograms suggests that there are significant outliers in the labs. Let’s look at how many observations lie above the 99th percentile of the data and the “outer fence” (defined as the 3rd quartile plus 3 time the interquartile range). We will then create new lab variables truncated from above at the outer fence and replot the histograms.

outer_fence <- function(v){
  q1 <- quantile(v, .25, na.rm = TRUE)
  q3 <- quantile(v, .75, na.rm = TRUE)
  iq <- (q3 - q1)
  return(as.numeric(q3 + 3 * iq))
}

format_percent <- function(x){
  paste0(formatC(100 * x, format = "f", digits = 1), "%")
}

train_data %>%
  select(one_of(lab_vars$var)) %>%
  pivot_longer(cols = lab_vars$var, names_to = "Lab") %>%
  filter(!is.na(value)) %>%
  group_by(Lab) %>%
  summarise(Maximum = max(value),
            `99%` = quantile(value, .99), 
            `Outer fence` = outer_fence(value),
            `% above outer fence` = format_percent(mean(value > outer_fence(value)))) %>%
  mutate(Lab = get_var_labs(Lab)) %>%
  kable() %>%
  kable_styling()
## `summarise()` ungrouping output (override with `.groups` argument)
Lab Maximum 99% Outer fence % above outer fence
Alanine aminotransferase (ALT) 3017.00 224.640000 130.000 3.4%
Aspartate aminotransferase (AST) 7000.01 336.000000 157.000 3.6%
Creatinine 27.35 10.878500 3.240 6.4%
C-reactive protein (CRP) 638.00 359.030000 458.000 0.2%
Fibrin D-Dimer 114475.00 19499.300000 4993.000 7.0%
Ferritin 100000.01 7747.270000 3646.875 3.4%
Lactate dehydrogenase (LDH) 14007.00 1262.520000 1049.500 1.9%
Lymphocyte count 120.35 3.736700 3.500 1.2%
Neutrophil count 82.40 18.900000 18.320 1.2%
Procalcitonin 753.66 30.122600 1.270 10.2%
Platelet count (PLT) 1213.00 527.830000 569.000 0.7%
Oxygen saturation 100.00 100.000000 110.000 0.0%
Troponin I 95.40 2.189095 0.170 8.1%
White blood cell count (WBC) 127.50 22.600000 21.700 1.2% 
# Truncate labs using outer fence
truncate_max <- function(v) outer_fence(v)

for (i in 1:length(lab_vars$var)){
  original_var <- lab_vars$var[i]
  truncated_var <- paste0(original_var, "_t")
  truncated_max_i <- truncate_max(train_data[[original_var]])
  train_data <- train_data %>% mutate(
    !!truncated_var := ifelse(get(original_var) > truncated_max_i, 
                              truncated_max_i, 
                              get(original_var))
    )
}

After truncating the labs, the probability distributions appear more reasonable.

lab_vars_t <- lab_vars %>%
  mutate(var = paste0(var, "_t"))
vars <- bind_rows(vars, lab_vars_t) 

continuous_vars_t <- bind_rows(
  demographic_continuous_vars,
  vital_continuous_vars,
  lab_vars_t
)

continuous_var_t_df <- pivot_continuous_longer(train_data, 
                                               vars = continuous_vars_t)
plot_hist(continuous_var_t_df)

3.3.1.2.4 Scatterplots
x_vars <- c("age")
y_vars <- c("bmi")
p <- vector(mode = "list", length = length(x_vars))
for (i in 1:length(x_vars)){
  p[[i]] <- ggplot(train_data, aes_string(x = x_vars[i], y = y_vars[i])) + 
  geom_point() + 
  #geom_smooth(method = "loess", formula = y ~ x) +
  xlab(get_var_labs(x_vars[i])) +
  ylab(get_var_labs(y_vars[i]))
}
do.call("grid.arrange", c(p, nrow = 1))
## Warning: Removed 1627 rows containing missing values (geom_point).

3.3.2 Outcomes

train_data %>%
  count(died) %>%
  mutate(died = ifelse(died == 1, "Yes", "No"),
         prop = n/sum(n)) %>%
  ggplot(aes(x = died, y = prop)) +
  geom_bar(stat = "identity") +
  geom_text(aes(label = formatC(prop, format = "f", digits = 2)),
            nudge_y = .03, size = 3) +
  xlab("Died") +
  ylab("Proportion")

3.3.2.1 Time to death

train_data %>%
  mutate(months_to_death = death_month - index_month) %>%
  filter(!is.na(months_to_death)) %>%
  group_by(months_to_death) %>%
  tally() %>%
  
  # Plot
  ggplot(aes(x = factor(months_to_death), y = n)) +
  geom_bar(stat = "identity", position = "dodge") +
  xlab("Months from index date to death") +
  ylab("Count")

3.4 Table 1

# Select variables for table 1
remove_varnames <- c("race", "ethnicity", "bmi_cat", "hypunc", "hypc", 
                     "diabunc", "diabwc")

all_varnames <- bind_rows(
  demographic_cat_vars,
  demographic_continuous_vars,
  comorbidity_vars,
  vital_cat_vars,
  vital_continuous_vars,
  lab_vars
) %>%
  filter(!var %in% remove_varnames) %>%
  pull(var)

categorical_varnames <- bind_rows(
  demographic_cat_vars,
  comorbidity_vars,
  vital_cat_vars
) %>%
  filter(!var %in% remove_varnames) %>%
  pull(var)

non_normal_varnames <- c(
  demographic_continuous_vars$var,
  vital_continuous_vars$var,
  lab_vars$var
)
  
## Create a TableOne object
table1 <- train_data %>%
  select(one_of(all_varnames, "died")) %>%
  rename_with(get_var_labs, .cols = all_of(all_varnames)) %>%
  rename(Survivor = died) %>%
  CreateTableOne(vars = get_var_labs(all_varnames), 
                 factorVars = get_var_labs(categorical_varnames), 
                 strata = "Survivor",  addOverall = TRUE, 
                 data = .)

## Print table 1
print(table1, nonnormal = get_var_labs(non_normal_varnames),  
      cramVars = c("Sex"),
      contDigits = 1, missing = TRUE, printToggle = FALSE) %>%
  set_colnames(c("Overall", "Survivor", "Non-survivor",
                "p", "Test", "Missing")) %>%
  kable() %>%
  kable_styling()
Overall Survivor Non-survivor p Test Missing
n 13666 11504 2162
Sex = Female/Male (%) 6565/7097 (48.1/51.9) 5638/5862 (49.0/51.0) 927/1235 (42.9/57.1) <0.001 0.0
Geographic division (%) <0.001 2.6
East North Central 4638 (34.9) 3965 (35.4) 673 (32.0)
East South Central 45 ( 0.3) 43 ( 0.4) 2 ( 0.1)
Middle Atlantic 4634 (34.8) 3842 (34.3) 792 (37.6)
Mountain 147 ( 1.1) 133 ( 1.2) 14 ( 0.7)
New England 1583 (11.9) 1272 (11.4) 311 (14.8)
Other 2 ( 0.0) 2 ( 0.0) 0 ( 0.0)
Pacific 511 ( 3.8) 438 ( 3.9) 73 ( 3.5)
South Atl/West South Crl 363 ( 2.7) 316 ( 2.8) 47 ( 2.2)
West North Central 1383 (10.4) 1189 (10.6) 194 ( 9.2)
Smoking (%) <0.001 25.6
Current 865 ( 8.5) 786 ( 9.0) 79 ( 5.3)
Never 6220 (61.1) 5459 (62.8) 761 (51.3)
Previous 3089 (30.4) 2447 (28.2) 642 (43.3)
Race/Ethnicity (%) <0.001 26.2
Non-Hispanic white 5646 (56.0) 4455 (53.1) 1191 (70.3)
Asian 362 ( 3.6) 307 ( 3.7) 55 ( 3.2)
Hispanic 536 ( 5.3) 481 ( 5.7) 55 ( 3.2)
Non-Hispanic black 3547 (35.2) 3153 (37.6) 394 (23.2)
Age (median [IQR]) 62.0 [49.0, 75.0] 59.0 [46.0, 71.0] 77.0 [67.0, 85.0] <0.001 nonnorm 0.0
Calendar time (median [IQR]) 46.0 [37.0, 63.0] 46.0 [37.0, 63.0] 45.0 [36.0, 59.0] <0.001 nonnorm 0.0
Acute myocardial infarction = Yes (%) 1534 (11.2) 1027 ( 8.9) 507 (23.5) <0.001 0.0
Congestive heart failure = Yes (%) 2323 (17.0) 1603 (13.9) 720 (33.3) <0.001 0.0
Peripheral vascular disease = Yes (%) 1671 (12.2) 1176 (10.2) 495 (22.9) <0.001 0.0
Cerebrovascular disease = Yes (%) 1439 (10.5) 1023 ( 8.9) 416 (19.2) <0.001 0.0
Dementia = Yes (%) 1394 (10.2) 854 ( 7.4) 540 (25.0) <0.001 0.0
Chronic pulmonary disease = Yes (%) 3622 (26.5) 2928 (25.5) 694 (32.1) <0.001 0.0
Rheumatoid disease = Yes (%) 398 ( 2.9) 315 ( 2.7) 83 ( 3.8) 0.006 0.0
Peptic ulcer disease = Yes (%) 206 ( 1.5) 160 ( 1.4) 46 ( 2.1) 0.013 0.0
Mild liver disease = Yes (%) 879 ( 6.4) 711 ( 6.2) 168 ( 7.8) 0.007 0.0
Hemiplegia or paraplegia = Yes (%) 330 ( 2.4) 228 ( 2.0) 102 ( 4.7) <0.001 0.0
Renal disease = Yes (%) 2833 (20.7) 1984 (17.2) 849 (39.3) <0.001 0.0
Cancer = Yes (%) 1677 (12.3) 1282 (11.1) 395 (18.3) <0.001 0.0
Moderate/severe liver disease = Yes (%) 128 ( 0.9) 88 ( 0.8) 40 ( 1.9) <0.001 0.0
Metastatic cancer = Yes (%) 277 ( 2.0) 188 ( 1.6) 89 ( 4.1) <0.001 0.0
AIDS/HIV = Yes (%) 100 ( 0.7) 88 ( 0.8) 12 ( 0.6) 0.361 0.0
Diabetes = Yes (%) 4612 (33.7) 3669 (31.9) 943 (43.6) <0.001 0.0
Hypertension = Yes (%) 8000 (58.5) 6331 (55.0) 1669 (77.2) <0.001 0.0
Body Mass Index (BMI) (median [IQR]) 29.7 [25.5, 35.1] 30.0 [25.8, 35.4] 28.1 [24.0, 33.5] <0.001 nonnorm 11.9
Diastolic blood pressure (median [IQR]) 73.0 [65.5, 80.5] 74.0 [66.6, 81.0] 68.0 [60.0, 75.5] <0.001 nonnorm 3.1
Systolic blood pressure (median [IQR]) 126.0 [115.0, 139.0] 127.0 [116.0, 139.0] 122.0 [109.0, 136.5] <0.001 nonnorm 3.2
Heart rate (median [IQR]) 87.5 [77.5, 98.0] 87.0 [77.5, 98.0] 89.0 [77.5, 102.0] <0.001 nonnorm 3.1
Respiration rate (median [IQR]) 20.0 [18.0, 22.0] 19.5 [18.0, 21.0] 22.0 [19.0, 26.0] <0.001 nonnorm 3.9
Temperature (median [IQR]) 37.0 [36.7, 37.4] 37.0 [36.7, 37.4] 37.1 [36.7, 37.6] <0.001 nonnorm 3.1
Alanine aminotransferase (ALT) (median [IQR]) 28.0 [18.0, 46.0] 28.0 [18.0, 46.0] 27.0 [18.0, 44.0] 0.099 nonnorm 20.1
Aspartate aminotransferase (AST) (median [IQR]) 37.0 [25.0, 58.0] 35.0 [25.0, 54.5] 46.0 [30.0, 73.0] <0.001 nonnorm 21.1
C-reactive protein (CRP) (median [IQR]) 79.1 [34.0, 140.0] 72.2 [30.0, 130.0] 116.0 [63.0, 184.0] <0.001 nonnorm 38.7
Creatinine (median [IQR]) 1.0 [0.8, 1.4] 1.0 [0.8, 1.3] 1.3 [1.0, 2.1] <0.001 nonnorm 10.4
Ferritin (median [IQR]) 510.0 [224.4, 1080.0] 470.0 [207.1, 992.0] 747.0 [320.0, 1500.0] <0.001 nonnorm 43.7
Fibrin D-Dimer (median [IQR]) 750.0 [390.0, 1540.8] 695.0 [370.0, 1346.5] 1345.0 [668.2, 3315.0] <0.001 nonnorm 90.4
Lactate dehydrogenase (LDH) (median [IQR]) 321.0 [238.0, 440.9] 308.0 [232.0, 415.0] 404.0 [284.0, 557.0] <0.001 nonnorm 45.3
Lymphocyte count (median [IQR]) 1.0 [0.7, 1.4] 1.0 [0.7, 1.4] 0.8 [0.5, 1.1] <0.001 nonnorm 11.2
Neutrophil count (median [IQR]) 4.9 [3.4, 7.1] 4.7 [3.2, 6.7] 6.1 [4.1, 9.2] <0.001 nonnorm 11.3
Oxygen saturation (median [IQR]) 96.0 [94.0, 98.0] 96.0 [94.5, 98.0] 95.0 [93.0, 97.0] <0.001 nonnorm 3.9
Procalcitonin (median [IQR]) 0.1 [0.1, 0.4] 0.1 [0.1, 0.3] 0.3 [0.1, 1.0] <0.001 nonnorm 49.3
Troponin I (median [IQR]) 0.0 [0.0, 0.0] 0.0 [0.0, 0.0] 0.0 [0.0, 0.1] <0.001 nonnorm 41.2
Platelet count (PLT) (median [IQR]) 202.0 [157.0, 260.0] 205.0 [160.0, 262.0] 187.8 [143.9, 245.0] <0.001 nonnorm 9.9
White blood cell count (WBC) (median [IQR]) 6.7 [4.9, 9.1] 6.5 [4.8, 8.7] 7.7 [5.6, 11.1] <0.001 nonnorm 9.7 
#summary(table1)

3.5 Transformations of continuous variables

The functional form of the relationship between mortality and the continuous variables is assessed using a series of univariate fits. We mainly rely on visual inspection of the graphs but also report the Bayesian information criterion (BIC).

fit_univariate_logit <- function(var, data){
  make_f <- function(rhs){
    as.formula(paste("died", rhs, sep =" ~ "))
  }
  fit_logit <- function(f, data){
    glm(f, family = "binomial", data = data)
  }
  
  list(
    `Linear` = fit_logit(make_f(var), data),
    `Spline 3 knots` = fit_logit(make_f(sprintf("ns(%s, 5)", var)), data),
    `Spline 4 knots` = fit_logit(make_f(sprintf("ns(%s, 6)", var)), data),
    `Spline 5 knots` = fit_logit(make_f(sprintf("ns(%s, 7)", var)), data)
  )
}

predict_univariate_logit <- function(models, var, var_values, type = "response"){
  newdata <- data.frame(var = var_values)
  colnames(newdata) <- var
  map_dfc(models,
         function(x) predict(x, newdata = newdata, type = type)) %>%
  mutate(var = var_values) %>%
  pivot_longer(cols = names(models),
               names_to = "Model",
               values_to = "y")
}

midpoint <- function(x, digits = 2){
  lower <- as.numeric(gsub(",.*", "", gsub("\\(|\\[|\\)|\\]", "", x)))
  upper <- as.numeric(gsub(".*,", "", gsub("\\(|\\[|\\)|\\]", "", x)))
  return(round(lower+(upper-lower)/2, digits))
}

bin_y <- function(var, var_values){
  data <- train_data[, c(var, "died")] %>%
    filter(!is.na(get(var)))
  data <- data %>%
    mutate(x_cat = cut(get(var), breaks = 20),
           x_midpoint = midpoint(x_cat)) %>%
    group_by(x_midpoint) %>%
    summarise(y = mean(died),
              n = n())
  colnames(data)[1] <- var
  return(data)
}

plot_univariate_logit <- function(models, var, var_values, var_lab = "Variable",
                                  type = "response", ylab = "Probability of death"){
  # Plotting data
  predicted_probs <- predict_univariate_logit(models, var, var_values, type = type)
  ylab <- switch(type,
                 "link" = "Log odds",
                 "response" = "Probability of death",
                 stop("Type must be 'link' or 'response'")
  )
  binned_y <- bin_y(var, var_values)
  if (type == "link"){
    binned_y$y <- ifelse(binned_y$y == 0, .001, binned_y$y)
    binned_y$y <- ifelse(binned_y$y == 1, .99, binned_y$y)
    binned_y$y <- qlogis(binned_y$y)
  }
  
  # Plotting scales
  y_min <- min(c(binned_y$y, predicted_probs$y))
  y_max <- max(c(binned_y$y, predicted_probs$y))
  size_breaks <- seq(min(binned_y$n), max(binned_y$n),
                     length.out = 6)
  
  # Plot
  ggplot(predicted_probs,
       aes(x = var, y = y)) +
    geom_line() +
    geom_point(data = binned_y, aes_string(x = var, y = "y", size = "n")) +
    facet_wrap(~Model, ncol = 2) +
    xlab(var_lab) +
    ylab(ylab) +
    ylim(floor(y_min), ceiling(y_max)) +
    scale_size(name = "Sample size", range = c(0.3, 3), 
               breaks = round(size_breaks, 0)) 
}

make_seq <- function(var){
  var_min <- min(train_data[[var]], na.rm = TRUE)
  var_max <- max(train_data[[var]], na.rm = TRUE)
  seq(var_min, var_max, length.out = 100)
}

evaluate_univariate_logit <- function(var, print = TRUE){
  var_values = make_seq(var)
  var_lab = get_var_labs(var)
  
  # Do evaluations
  fits <- fit_univariate_logit(var, data = train_data)
  p_link <- plot_univariate_logit(fits, var, var_values, var_lab, type = "link")
  p_probs <- plot_univariate_logit(fits, var, var_values, var_lab, type = "response")
  bic <- sapply(fits, BIC)
  
  # Print and return
  if (print){
    print(p_link)
    print(p_probs)
    print(bic)
  }
  return(list(fits = fits, p_link = p_link, p_probs = p_probs,
              bic = bic))
}

3.5.1 Demographics

3.5.1.1 Age

uv_age <- evaluate_univariate_logit("age")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10152.05       10178.63       10188.22       10197.59

3.5.1.2 Calendar time

uv_calendar_time <- evaluate_univariate_logit("calendar_time")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11926.53       11947.74       11956.52       11965.13

3.5.2 Vitals

3.5.2.1 Body Mass Index

uv_bmi <- evaluate_univariate_logit("bmi")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10631.88       10625.62       10634.47       10640.01

3.5.2.2 Diastolic blood pressure

uv_dbp <- evaluate_univariate_logit("dbp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11075.50       11067.63       11077.10       11086.28

3.5.2.3 Systolic blood pressure

uv_sbp <- evaluate_univariate_logit("sbp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11464.70       11247.39       11256.25       11265.91

3.5.2.4 Heart rate

uv_hr <- evaluate_univariate_logit("hr")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11501.36       11405.73       11413.55       11421.89

3.5.2.5 Respiration rate

uv_resp <- evaluate_univariate_logit("resp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10738.21       10504.93       10513.53       10521.69

3.5.2.6 Temperature

uv_temp <- evaluate_univariate_logit("temp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11595.41       11353.97       11361.13       11368.32

3.5.3 Labs

3.5.3.1 Alanine aminotransferase (U/L)

uv_alt <- evaluate_univariate_logit("alt")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10127.24       10157.07       10164.25       10170.19
uv_alt_t <- evaluate_univariate_logit("alt_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10125.03       10158.67       10168.22       10176.69

3.5.3.2 Alanine aminotransferase (U/L)

uv_ast <- evaluate_univariate_logit("ast")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       9996.867       9838.208       9845.513       9853.245
uv_ast_t <- evaluate_univariate_logit("ast_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       9849.657       9841.263       9850.436       9859.691

3.5.3.3 C-reactive protein (mg/L)

uv_crp <- evaluate_univariate_logit("crp")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7610.895       7598.575       7606.895       7613.547
uv_crp_t <- evaluate_univariate_logit("crp_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7609.220       7598.523       7606.862       7613.446

3.5.3.4 Creatinine (mg/dL)

uv_creatinine <- evaluate_univariate_logit("creatinine")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10991.89       10511.32       10499.98       10500.91
uv_creatinine_t <- evaluate_univariate_logit("creatinine_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10599.56       10477.76       10485.67       10492.08

3.5.3.5 Ferritin (ng/mL)

uv_ferritin <- evaluate_univariate_logit("ferritin")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7311.431       7200.894       7211.518       7217.965
uv_ferritin_t <- evaluate_univariate_logit("ferritin_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7216.741       7203.104       7214.089       7219.304

3.5.3.6 Fibrin D-Dimer (ng/mL)

uv_d_dimer <- evaluate_univariate_logit("d_dimer")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       1201.619       1189.305       1193.129       1197.296
uv_d_dimer_t <- evaluate_univariate_logit("d_dimer_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       1154.320       1177.934       1185.052       1192.091

3.5.3.7 Lactate dehydrogenase (U/L)

uv_ldh <- evaluate_univariate_logit("ldh")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6918.661       6729.046       6731.922       6737.658
uv_ldh_t <- evaluate_univariate_logit("ldh_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6724.449       6713.154       6722.186       6730.649

3.5.3.8 Lymphocyte count (10^3/uL)

uv_lymphocyte <- evaluate_univariate_logit("lymphocyte")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11015.74       10671.85       10679.94       10688.67
uv_lymphocyte_t <- evaluate_univariate_logit("lymphocyte_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10834.08       10664.96       10672.87       10681.37

3.5.3.9 Neutrophil count (10^3/uL)

uv_neutrophil <- evaluate_univariate_logit("neutrophil")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10736.10       10717.76       10718.11       10725.33
uv_neutrophil_t <- evaluate_univariate_logit("neutrophil_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10704.88       10705.77       10711.41       10719.06

3.5.3.10 Oxygen saturation (%)

uv_spo2 <- evaluate_univariate_logit("spo2")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11140.41       11107.89       11117.45       11127.13

3.5.3.11 Procalcitonin (ng/mL)

uv_pct <- evaluate_univariate_logit("pct")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6956.536       6499.081       6498.854       6504.554
uv_pct_t <- evaluate_univariate_logit("pct_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       6566.560       6495.053       6497.433       6506.346

3.5.3.12 Troponin I (ng/mL)

uv_tni <- evaluate_univariate_logit("tni")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7754.330       7045.431       7050.912       7059.858
uv_tni_t <- evaluate_univariate_logit("tni_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       7077.283       6954.025       6962.474       6970.854

3.5.3.13 Platelet count (10^3/uL)

uv_plt <- evaluate_univariate_logit("plt")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11101.79       11079.13       11086.73       11095.29
uv_plt_t <- evaluate_univariate_logit("plt_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       11098.10       11078.63       11086.54       11095.09

3.5.3.14 White blood cell count (10^3/uL)

uv_wbc <- evaluate_univariate_logit("wbc")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10985.17       10939.94       10944.21       10950.75
uv_wbc_t <- evaluate_univariate_logit("wbc_t")

##         Linear Spline 3 knots Spline 4 knots Spline 5 knots 
##       10932.45       10925.13       10932.84       10939.63

4 Data preprocessing

4.1 Candidate variables for modeling

Let’s specify variables that will be candidates for our model and used during variable selection. We will combine race and ethnicity into a single variable, but will wait to do this until after multiple imputation (see next section).

demographics_to_include <- c("age", "sex", "race", "ethnicity", "division",
                             "smoke", "calendar_time")
comorbidities_to_include <- comorbidity_vars %>%
  filter(!var %in% c("diabunc", "diabwc", "hypunc", "hypc")) %>%
  pull(var) %>%
  c("diab", "hyp")
vitals_to_include <- c("bmi", "temp", "hr", "resp", "sbp")
labs_to_include <- c("spo2", "crp_t", "tni_t", "ast_t", "ferritin_t",
                     "creatinine_t", "ldh_t", "lymphocyte_t", "neutrophil_t",
                     "plt_t", "wbc_t")
vars <- vars %>%
  mutate(include = ifelse(var %in% c(demographics_to_include, 
                                     comorbidities_to_include,
                                     vitals_to_include, 
                                     labs_to_include),
                          1, 0))
get_included_vars <- function(){
  vars[vars$include == 1, ]$var
}

make_rhs <- function(vars){
  as.formula(paste0("~", paste(vars, collapse = " + ")))
}
candidate_model_rhs <- make_rhs(get_included_vars())

4.2 Missing data imputation

Note that there is no “Other” geographic region as all 9 of the US geographic divisions are included in the data. We will consequently recode this category to missing.

train_data <- train_data %>%
  mutate(division = ifelse(division == "Other", NA, division),
         division = ifelse(division %in% c("East South Central", "Mountain"),
                           "Other", division))

Let’s re-level them so that we have preferred reference categories.

train_data <- train_data %>% mutate(
  sex = relevel(factor(sex), ref = "Male"),
  division = relevel(factor(division), ref = "Pacific"),
  smoke = relevel(factor(smoke), ref = "Never"),
  bmi_cat = relevel(factor(bmi_cat), ref = "Normal")
)

We can now multiply impute data use MICE.

# Run MICE algorithm
mice_out <- train_data %>%
  select(c(one_of(get_included_vars()), "died")) %>%
  mutate_if(is.character, as.factor) %>%
  mice(m = n_imputations, maxit = 5) 
## 
##  iter imp variable
##   1   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   1   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   2   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   2   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   3   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   3   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   4   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   4   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   5   1  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
##   5   2  sex  race  ethnicity  division  smoke  bmi  sbp  hr  resp  temp  spo2  ast_t  crp_t  creatinine_t  ferritin_t  ldh_t  lymphocyte_t  neutrophil_t  tni_t  plt_t  wbc_t
# Append datasets and add outcome
mi_df <- complete(mice_out, action = "long", include = TRUE) %>%
  as_tibble() %>%
  mutate(died = rep(train_data$died, mice_out$m + 1))

# To compare MICE to aregImpute
# areg_out <-  aregImpute(update.formula(candidate_model_rhs, ~.), 
#                                        n.impute = 2, data = train_data)

4.3 Imputation diagnostics

The distributions of the imputed and observed data are compared as a diagnostic for the imputation. They look pretty similar suggesting that there is nothing terribly wrong with the imputation.

make_imp_df <- function(object){
  # Get imputations
  if (inherits(object, "mids")){
    imp <- object$imp
  } else{ # aregImpute
   imp <- object$imputed 
   for (i in 1:length(imp)){
     cat_levels_i <- object$cat.levels[[i]]
     if (!is.null(cat_levels_i) && !is.null(imp[[i]])){
       levels <- sort(unique(c(imp[[i]])))
       imp[[i]] <- apply(imp[[i]], 
                         2, 
                         function(x) factor(x, levels = levels,
                                            labels = cat_levels_i))
     }
   }
  }
  
  # Create list of data frames
  is_numeric <- sapply(imp, function (x) is.numeric(x[, 1]))
  continuous_df <- vector(mode = "list", length = sum(is_numeric))
  cat_df <- vector(mode = "list", length = sum(!is_numeric))
  continuous_cntr <- 1
  cat_cntr <- 1
  for (i in 1:length(imp)){
    if(!is.null(nrow(imp[[i]])) && nrow(imp[[i]]) > 0 ){
        imp_i_df <- data.frame(var = names(imp)[i],
                               imp = rep(1:ncol(imp[[i]]), each = nrow(imp[[i]])),
                               value = c(as.matrix(imp[[i]]))) %>% 
          as_tibble()
        
    } else{
      imp_i_df <- NULL
    }
    if (is_numeric[i]){
      continuous_df[[continuous_cntr]] <- imp_i_df
      continuous_cntr <- continuous_cntr + 1
    } else{
      cat_df[[cat_cntr]] <- imp_i_df
      cat_cntr <- cat_cntr + 1
    }
  } 
  
  # Row bind data frames
  continuous_df = bind_rows(continuous_df) %>%
    mutate(obs = "Imputed",
           varlab = get_var_labs(var)) 
  
  cat_df = bind_rows(cat_df) %>%
    mutate(obs = "Imputed",
           varlab = get_var_labs(var)) 
              
  # Return
  return(list(continuous = continuous_df,
              cat = cat_df))
}
imp_df <- make_imp_df(mice_out)
#imp_df <- make_imp_df(areg_out)
# Plot continuous variables
## Data for plotting
obsimp_df_continuous <- bind_rows(
  imp_df$continuous,
  continuous_var_df %>%
    select(var, value, varlab) %>%
    mutate(imp = 0, obs = "Observed")
) %>%
  mutate(imp = ifelse(imp == 0, "Observed", paste0("Imputation ", imp))) %>%
  filter(var %in% unique(imp_df$continuous$var))
  
## Plot
ggplot(obsimp_df_continuous,
       aes(x = value, col = imp)) +
  geom_density(position = "jitter") +
  facet_wrap(~varlab, scales = "free", ncol = 3) +
  xlab("") + ylab("Density") +
  scale_color_discrete(name = "") +
  theme(legend.position = "bottom")

# Plot categorical variables
## Data for plotting
obsimp_df_cat <- 
  bind_rows(
    imp_df$cat %>%
      group_by(var, varlab, value, imp) %>%
      summarise(n = n()) %>%
      group_by(var, varlab, imp) %>%
      mutate(freq = n / sum(n)),
    cat_var_df %>%
      select(var, value, varlab, n, freq) %>%
      mutate(imp = 0, obs = "Observed")
  ) %>%
    mutate(imp = ifelse(imp == 0, "Observed", paste0("Imputation ", imp))) %>%
    filter(var %in% unique(imp_df$cat$var))
## `summarise()` regrouping output by 'var', 'varlab', 'value' (override with `.groups` argument)
# Plot
ggplot(obsimp_df_cat, 
       aes(x = value, y = freq, fill = imp)) +
  geom_bar(position = "dodge", stat = "identity") +
  facet_wrap(~varlab, scales = "free_x") +
  scale_fill_discrete(name = "") +
  xlab("") +
  ylab("Proportion") +
  theme(legend.position = "bottom",
        axis.text.x = element_text(angle = 45, vjust = 1, hjust = 1))

4.4 Recoding categorical variables

We create the race/ethnicity variable after imputation of missing values of race and ethnicity. We will also combine the diabetes and hypertension variables.

mi_df <- mi_df %>% 
  add_race_ethnicity() 

prop.table(table(mi_df %>% filter(.imp == 0) %>% pull(diab)))
## 
##        No       Yes 
## 0.6625201 0.3374799
prop.table(table(mi_df %>% filter(.imp == 0) %>% pull(hyp)))
## 
##        No       Yes 
## 0.4146056 0.5853944

Now lets use the combined race and ethnicity category in our model.

vars <- vars %>%
  mutate(include = case_when(
      var == "race_ethnicity" ~ 1,
      var == "race" ~ 0,
      var == "ethnicity" ~ 0,
      TRUE ~ include
    )
  )
  
candidate_model_rhs <- make_rhs(get_included_vars())

Finally, we will (i) add the new race/ethnicity variable to the “MICE” object and (ii) create a list of imputed datasets for analysis.

mice_out <- as.mids(mi_df)
mi_list <- mi_df %>% 
  filter(.imp > 0) %>% 
  split(list(.$.imp))

5 Variable selection

The continuous variables in our model are transformed using restricted cubic splines with 3 knots.

candidate_model_rhs <- candidate_model_rhs %>%
  update.formula( ~. + rcs(age, 3) - age +
                       rcs(calendar_time, 3) - calendar_time +
                       rcs(bmi, 3) - bmi +
                       rcs(temp, 3) - temp +
                       rcs(hr, 3) - hr +
                       rcs(resp, 3) - resp +
                       rcs(sbp, 3) - sbp +
                       rcs(spo2, 3) - spo2 +
                       rcs(crp_t, 3) - crp_t +
                       rcs(tni_t, 3) - tni_t +
                       rcs(ast_t, 3) - ast_t +
                       rcs(creatinine_t, 3) - creatinine_t +
                       rcs(ferritin_t, 3) - ferritin_t +
                       rcs(ldh_t, 3) - ldh_t,
                       rcs(lymphocyte_t, 3) - lymphocyte_t +
                       rcs(neutrophil_t, 3) - neutrophil_t +
                       rcs(plt_t, 3) - plt_t +
                       rcs(wbc_t, 3) - wbc_t)

With oem we need to create an x matrix since there is no formula interface.

rename_rcs <- function(v){
  rcs_ind <- grep("rcs", v)
  v[rcs_ind] <- sub("rcs.*)", "", v[rcs_ind])
  return(v)
}

rename_terms <- function(v){
  v <- gsub(" ", "_", v)
  v <- gsub("-", "", v)
  v <- gsub("/", "", v)
  v <- rename_rcs(v)
  return(v)
}

make_x <- function(data){
  x <- model.matrix(candidate_model_rhs, data)
  assign <- attr(x, "assign")
  colnames(x) <- rename_terms(colnames(x))
  x <- x[, -1]
  attr(x, "assign") <- assign[-1]
  return(x)
}

# List of x and y for each imputed dataset
x <- mi_list %>% map(make_x)
y <- mi_list %>% map(function(x) x[["died"]])

To make the graphs look nice, we will import labels for the model terms.

terms <- read.csv("risk-factors-terms.csv") %>%
  left_join(vars[, c("var", "group")], by = "var")

get_term_labs <- function(v, term_name = "term"){
  terms$termlab[match(v, terms[[term_name]])]
}

match_terms_to_vars <- function(t){
  terms$var[match(t, terms$term)]
}

Finally, we run the grouped lasso.

# Number of folds for cross-validation
n_folds <- 10

# Threshold for variable inclusion
inclusion_threshold <- 0.9

# Matrix to store inclusion results
inclusion_sim <- matrix(0,  ncol = ncol(x[[1]]) + 1, 
                        nrow = n_rep * n_imputations)

# Groups
groups <- attr(x[[1]], "assign") 

# Convenience function to extract coefficients from Group-Lasso
coef_cv_oem <- function(object){
  coef <- object$oem.fit$beta$grp.lasso
  lse_ind <- which(object$lambda[[1]] == object$lambda.1se.models)
  return(coef[, lse_ind])
}

# Variable selection via Group-Lasso
cntr <- 1
for (i in 1:n_imputations){
  for (j in 1:n_rep){
    
    # Cross-validation
    cv_fit <-  cv.oem(x = x[[i]], y = y[[i]], 
                      penalty = "grp.lasso", 
                      groups = groups, 
                      family = "binomial",
                      type.measure = "deviance",
                      nfolds = n_folds
                      )
  
    # Count nonzero coefficients 
    inclusion_sim[cntr, which(coef_cv_oem(cv_fit) != 0)] <- 1
    
    # Iterate
    cntr <- cntr + 1
  } # End repeated CV loop
} # End imputation loop

Let’s check the inclusion probabilities from the above repeated cross-validation steps.

# Percentage of simulations each term is included
inclusion_sim <- inclusion_sim[, -1] # Remove intercept
colnames(inclusion_sim) = colnames(x[[1]])

inclusion_summary <- tibble(term = colnames(inclusion_sim), 
                            prob = apply(inclusion_sim, 2, mean)) 
model_terms <- inclusion_summary %>%
  filter(prob >= inclusion_threshold) %>%
  pull(term)

# Percentage of simulations each variable is included
inclusion_summary <- inclusion_summary %>%
  mutate(var = match_terms_to_vars(term)) %>%
  mutate(varlab = get_var_labs(var)) %>% 
  distinct(prob, var, varlab)

# Plot
ggplot(inclusion_summary,
       aes(x = reorder(varlab, prob), y = prob)) +
  geom_bar(stat = "identity") + 
  geom_hline(yintercept = inclusion_threshold, linetype = "dotted", 
                color = "red", size = 1) +
  ylab("Probability of inclusion") +
  coord_flip() +
  theme(axis.title.y = element_blank())

6 Model interpretation

6.1 Model terms and data

Variables are included in the model based on the group Lasso simulation implemented above.

vars_to_exclude <- inclusion_summary %>%
  filter(prob < inclusion_threshold) %>%
  pull(var)

remove_terms_from_rhs <- function(f, vars_to_exclude){
  # First convert formula to string separated by +
  f_string <- Reduce(paste, deparse(f))
  f_string <- gsub("~", "", f_string)
  f_string <- gsub(" ", "", f_string)
  
  # Then convert string to vector
  f_vec <-   unlist(strsplit(f_string, "\\+"))
  pattern_to_exclude <- paste(vars_to_exclude, collapse = "|")
  f_vec <- f_vec[!grepl(pattern_to_exclude, f_vec)]
  
  # Convert string back to formula
  f_new <- paste0("~", paste(f_vec, collapse = " + "))
  return(as.formula(f_new))
}

model_rhs <- remove_terms_from_rhs(candidate_model_rhs, vars_to_exclude)
label(mi_df) <- map(colnames(mi_df), 
                    function(x) label(mi_df[, x]) <- get_var_labs(x))
dd <- datadist(mi_df, adjto.cat = "first")
options(datadist = "dd")

6.2 Fit

As described in the paper, four models are fit that include the following predictors: (i) age only, (ii) all demographics (and calendar time), and (iii) demographics (and calendar time) and comorbidities, and (iv) all variables selected by the grouped lasso.

f_logit_all <- update.formula(model_rhs, died ~ .)
# The four models
lrm_names <- c("Age only",
                "All demographics",
                "Demographics and comorbidities",
                "All variables")

## (1): fit_age: Only includes age
f_logit_age <- died ~ age 
lrm_fit_age <- fit.mult.impute(f_logit_age, fitter = lrm, xtrans = mice_out, pr = FALSE,
                                x = TRUE, y = TRUE)

## (2): fit_d: All demographics including age
f_logit_d <- update.formula(f_logit_age, ~. + sex + rcs(calendar_time, 3) + 
                              race_ethnicity + division)
lrm_fit_d <- fit.mult.impute(f_logit_d, fitter = lrm, xtrans = mice_out, pr = FALSE,
                             x = TRUE, y = TRUE)

## (3): fit_dc: Demographics and comorbidities
c_vars <- vars %>%
  filter(group == "Comorbidities" & include == 1 & !var %in% vars_to_exclude) %>%
  pull(var)
f_logit_dc <- update.formula(f_logit_d, 
                             as.formula(paste0("~.+", paste(c_vars, collapse = "+"))))
lrm_fit_dc <- fit.mult.impute(f_logit_dc, fitter = lrm, xtrans = mice_out, pr = FALSE,
                              x = TRUE, y = TRUE)

## (4): fit: The main model including demographics, comorbidities, vitals, and labs
lrm_fit_all <- fit.mult.impute(f_logit_all, fitter = lrm, xtrans = mice_out, 
                               pr = FALSE, x = TRUE, y = TRUE)

# Note that we can also estimate the models with stats::glm
glm_fits_all <- mi_list %>%
  map(function (x) glm(f_logit_all, data = x, family = "binomial")) 
glm_fit_all <- glm_fits_all %>% pool()

We will print the results of the full model for the interested reader

lrm_fit_all
## Logistic Regression Model
##  
##  fit.mult.impute(formula = f_logit_all, fitter = lrm, xtrans = mice_out, 
##      pr = FALSE, x = TRUE, y = TRUE)
##  
##                         Model Likelihood    Discrimination    Rank Discrim.    
##                               Ratio Test           Indexes          Indexes    
##  Obs         13666    LR chi2    4231.45    R2       0.457    C       0.890    
##   0          11504    d.f.            59    g        2.402    Dxy     0.780    
##   1           2162    Pr(> chi2) <0.0001    gr      11.047    gamma   0.781    
##  max |deriv| 2e-09                          gp       0.205    tau-a   0.208    
##                                             Brier    0.087                     
##  
##                                    Coef     S.E.    Wald Z Pr(>|Z|)
##  Intercept                          13.6351  2.6127  5.22  <0.0001 
##  sex=Female                         -0.0229  0.0667 -0.34  0.7307  
##  division=East North Central        -0.3840  0.1941 -1.98  0.0479  
##  division=Middle Atlantic           -0.2833  0.1728 -1.64  0.1011  
##  division=New England               -0.1864  0.1980 -0.94  0.3466  
##  division=Other                     -0.8502  0.3351 -2.54  0.0112  
##  division=South Atl/West South Crl  -0.4907  0.2631 -1.86  0.0622  
##  division=West North Central        -0.4450  0.2095 -2.12  0.0337  
##  smoke=Current                      -0.0405  0.1332 -0.30  0.7613  
##  smoke=Previous                      0.1762  0.0908  1.94  0.0523  
##  race_ethnicity=Asian               -0.1019  0.1903 -0.54  0.5924  
##  race_ethnicity=Hispanic            -0.1858  0.1254 -1.48  0.1386  
##  race_ethnicity=Non-Hispanic black  -0.5651  0.0875 -6.46  <0.0001 
##  ami=Yes                             0.0366  0.0851  0.43  0.6671  
##  chf=Yes                             0.1837  0.0769  2.39  0.0169  
##  pvd=Yes                             0.0599  0.0821  0.73  0.4660  
##  cevd=Yes                            0.0943  0.0885  1.07  0.2862  
##  dementia=Yes                        0.3787  0.0834  4.54  <0.0001 
##  copd=Yes                            0.0498  0.0700  0.71  0.4764  
##  rheumd=Yes                          0.0571  0.1586  0.36  0.7189  
##  mld=Yes                             0.0839  0.1330  0.63  0.5281  
##  hp=Yes                              0.7286  0.1583  4.60  <0.0001 
##  rend=Yes                            0.1088  0.0823  1.32  0.1862  
##  canc=Yes                            0.0325  0.0860  0.38  0.7053  
##  msld=Yes                            0.7995  0.2551  3.13  0.0017  
##  metacanc=Yes                        0.8501  0.1826  4.65  <0.0001 
##  aids=Yes                            0.6647  0.3651  1.82  0.0687  
##  diab=Yes                            0.1244  0.0664  1.87  0.0609  
##  hyp=Yes                             0.1104  0.0787  1.40  0.1606  
##  lymphocyte_t                       -0.2263  0.0561 -4.03  <0.0001 
##  plt_t                              -0.0025  0.0004 -6.15  <0.0001 
##  wbc_t                               0.0465  0.0112  4.14  <0.0001 
##  age                                 0.0672  0.0044 15.39  <0.0001 
##  age'                               -0.0046  0.0056 -0.82  0.4106  
##  calendar_time                      -0.0176  0.0047 -3.70  0.0002  
##  calendar_time'                      0.0079  0.0081  0.97  0.3334  
##  bmi                                -0.0254  0.0101 -2.53  0.0115  
##  bmi'                                0.0460  0.0119  3.88  0.0001  
##  temp                               -0.1885  0.0530 -3.55  0.0004  
##  temp'                               0.9287  0.0947  9.81  <0.0001 
##  hr                                  0.0020  0.0044  0.45  0.6526  
##  hr'                                 0.0109  0.0049  2.22  0.0265  
##  resp                                0.0415  0.0259  1.60  0.1090  
##  resp'                               0.0429  0.0250  1.72  0.0858  
##  sbp                                -0.0334  0.0036 -9.34  <0.0001 
##  sbp'                                0.0314  0.0043  7.34  <0.0001 
##  spo2                               -0.1232  0.0169 -7.27  <0.0001 
##  spo2'                               0.1289  0.0279  4.61  <0.0001 
##  crp_t                               0.0033  0.0018  1.84  0.0651  
##  crp_t'                             -0.0035  0.0020 -1.73  0.0838  
##  tni_t                              17.5611  3.6648  4.79  <0.0001 
##  tni_t'                            -47.8269 13.7956 -3.47  0.0005  
##  ast_t                               0.0081  0.0042  1.90  0.0573  
##  ast_t'                             -0.0155  0.0076 -2.04  0.0417  
##  creatinine_t                        0.2842  0.1299  2.19  0.0288  
##  creatinine_t'                      -0.1662  0.2269 -0.73  0.4638  
##  ferritin_t                          0.0002  0.0002  0.90  0.3685  
##  ferritin_t'                        -0.0003  0.0005 -0.72  0.4688  
##  ldh_t                               0.0050  0.0013  3.82  0.0001  
##  ldh_t'                             -0.0042  0.0018 -2.38  0.0172  
##

6.3 Collinearity

Let’s examine the extent to which our predictors are collinear: for categorical and continous variables, we use an anova model; for categorical and categorical variables, we use Cramer’s V; and for continous and continous variables, we use spearman correlation. (Note that this takes a long time to run and could probablty be made more efficient.)

## from https://stackoverflow.com/questions/52554336/plot-the-equivalent-of-correlation-matrix-for-factors-categorical-data-and-mi
mixed_assoc <- function(df, cor_method = "spearman", adjust_cramersv_bias = TRUE){
  df_comb <- expand.grid(names(df), names(df),  stringsAsFactors = F) %>% 
    set_names("X1", "X2")
  is_nominal <- function(x) inherits(x, c("factor", "character"))
  # https://community.rstudio.com/t/why-is-purr-is-numeric-deprecated/3559
  # https://github.com/r-lib/rlang/issues/781
  is_numeric <- function(x) { is.integer(x) || is_double(x)}
  f <- function(x_name, y_name) {
    x <-  pull(df, x_name)
    y <-  pull(df, y_name)
    result <- if(is_nominal(x) && is_nominal(y)){
        # use bias corrected cramersV as described in https://rdrr.io/cran/rcompanion/man/cramerV.html
        cv <- cramerV(as.character(x), as.character(y), 
                      bias.correct = adjust_cramersv_bias)
        data.frame(x_name, y_name, assoc = cv, type = "cramersV")
    } else if(is_numeric(x) && is_numeric(y)){
        correlation <- cor(x, y, method = cor_method, use = "complete.obs")
        data.frame(x_name, y_name, assoc = correlation, type = "correlation")
    } else if(is_numeric(x) && is_nominal(y)){
        # from https://stats.stackexchange.com/questions/119835/correlation-between-a-nominal-iv-and-a-continuous-dv-variable/124618#124618
        r_squared <- summary(lm(x ~ y))$r.squared
        data.frame(x_name, y_name, assoc = sqrt(r_squared), type = "anova")
    } else if(is_nominal(x) && is_numeric(y)){
        r_squared <- summary(lm(y ~ x))$r.squared
        data.frame(x_name, y_name, assoc = sqrt(r_squared), type = "anova")
    } else {
        warning(paste("unmatched column type combination: ", class(x), class(y)))
    }
    # finally add complete obs number and ratio to table
    result %>% 
      mutate(
        complete_obs_pairs = sum(!is.na(x) & !is.na(y)),
        complete_obs_ratio = complete_obs_pairs/length(x)) %>%
      rename(x = x_name, y = y_name)
  }
  # apply function to each variable combination
  map2_df(df_comb$X1, df_comb$X2, f)
}
# Create correlation matrix of associations
corr_mat <- mi_df %>% 
  filter(.imp == 1) %>% 
  select(any_of(get_included_vars())) %>%
  mixed_assoc() %>%
  select(x, y, assoc) %>%
  pivot_wider(names_from = y, values_from = assoc) %>%
  column_to_rownames("x") %>%
  as.matrix

# Make tile plot
m <- abs(corr_mat)
heatmap_df <- tibble(row = rownames(m)[row(m)], 
                     col = colnames(m)[col(m)], corr = c(m)) %>%
  mutate(row = get_var_labs(row),
         col = get_var_labs(col))
heatmap_df %>%
  ggplot(aes(x = row, y = col, fill = corr)) + 
  geom_tile() + 
  scale_fill_continuous("Correlation") +
  theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust = 1),
        axis.title = element_blank())

6.4 Variable importance

Variable importance is assessed with a Wald test using the rms::anova().

# Compute variable important with Wald chi-square
lrm_anova_all <- anova(lrm_fit_all)

# Plot the result
lrm_anova_all %>% 
  as_tibble() %>%
  mutate(var = gsub(" ", "", rownames(lrm_anova_all)),
         varlab = get_var_labs(var),
         value = as.double(`Chi-Square` - `d.f.`)) %>%
  filter(!var %in% c("TOTAL", "Nonlinear", "TOTALNONLINEAR")) %>%

  ggplot(aes(x = value, y = reorder(varlab, value))) +
  geom_point() +
  theme(axis.title.y = element_blank()) +
  xlab(expression(chi^2-df))

6.5 Summary of odds ratios

The model is summarized with odds ratios using rms::summary(). We will start by assessing the full model. Next, since labs and vitals may themselves be “caused” by comorbidities, we will see how the odds ratios for the comorbidities change after dropping labs and vitals from the model.

6.5.1 Full model

lrm_summary_all <- summary(lrm_fit_all)

# Odds ratios
format_or_range <- function(x, term){
  case_when(
    x < 10 ~ formatC(x, format = "f", digits = 2),
    term == "temp" ~ formatC(x, format = "f", digits = 1),
    TRUE ~ formatC(x, format = "f", digits = 0)
  )
}

make_tidy_or <- function(object, model_name = NULL){
  if (is.null(model_name)) model_name <- "Model"
  object %>%
    as.data.frame() %>%
    as_tibble() %>%
    mutate(term = rownames(object),
           High = format_or_range(High, term),
           Low = format_or_range(Low, term),
           termlab = get_term_labs(term, "term2"),
           termlab = ifelse(!is.na(`Diff.`), 
                            paste0(termlab, " - ", High, ":", Low),
                            termlab),
           or = exp(Effect),
           or_lower = as.double(exp(`Lower 0.95`)),
           or_upper = exp(`Upper 0.95`)) %>%
    filter(Type == 1) %>%
    select(term, termlab, or, or_lower, or_upper) %>%
    arrange(-or) %>%
    mutate(model = model_name)
}
lrm_or_all <- make_tidy_or(lrm_summary_all, "All variables") 

# Odds ratio plot
ggplot(lrm_or_all, 
       aes(x = or, y = reorder(termlab, or))) +
  geom_point() +
  geom_errorbarh(aes(xmax = or_upper, xmin = or_lower,
                     height = .2)) +
  geom_vline(xintercept = 1, linetype = "dashed", col = "grey") + 
  theme(axis.title.y = element_blank()) +
  xlab("Odds ratio")

6.5.2 Comparison of model with and without vitals + labs

lrm_summary_dc <- summary(lrm_fit_dc)
lrm_or_dc <- make_tidy_or(lrm_summary_dc, "Demographics + comorbidities")

# Odds ratio comparison plot
lrm_or_comp <- bind_rows(lrm_or_all, lrm_or_dc) %>%
  filter(term %in% 
           terms[terms$group %in% c("Demographics", "Comorbidities"), ]$term2) %>%
  mutate(termlab = factor(termlab),
         termlab = reorder(termlab, or, function (x) -mean(x)))

ggplot(lrm_or_comp, 
       aes(x = termlab, y = or, col = model)) +
  geom_point(position = position_dodge(width = 1)) +
  geom_errorbar(aes(ymax = or_upper, ymin = or_lower,
                     width = .2), position = position_dodge(width = 1)) +
  facet_wrap(~termlab, strip.position = "left", ncol = 1, scales = "free_y") + 
  geom_hline(yintercept = 1, linetype = "dashed") + 
  theme(axis.title.y = element_blank()) +
  scale_color_discrete(name = "Model") +
  theme(axis.text.y = element_blank(),
        axis.ticks.y = element_blank(),
        strip.text.y.left = element_text(hjust = 0, vjust = 1, 
                                         angle = 0, size = 8),
        legend.position = "bottom") +
  ylab("Odds ratio") +
  coord_flip()

6.6 Predicted log odds

One limitation of the odds ratio plots is that it is hard to examine the potentially non-linear relationships between mortality and the continuous variable. The rms::Predict() function is especially helpul in this regard. We use it to vary all of the predictors and plot predicted log odds across the different values of each predictor.

Each prediction is made with all other variables at their “Adjust to” value as specified with datadist() above.

t(dd$limits) %>%
  as_tibble() %>%
  mutate(Variable = get_var_labs(colnames(dd$limits))) %>%
  relocate(Variable, .before = "Low:effect") %>%
  filter(!is.na(Variable)) %>%
  arrange(Variable) %>%
  kable() %>%
  kable_styling()
Variable Low:effect Adjust to High:effect Low:prediction High:prediction Low High
Acute myocardial infarction NA No NA No Yes No Yes
Age 49 62 75 18 89 18 89
AIDS/HIV NA No NA No Yes No Yes
Aspartate aminotransferase (AST) 25 36 57 6 157 4 157
Body Mass Index (BMI) 25.54000 29.73000 35.16000 13.32705 76.58000 11.84000 149.50000
C-reactive protein (CRP) 29.0 72.0 133.0 0.2 458.0 0.0 458.0
Calendar time 37 46 63 0 90 0 90
Cancer NA No NA No Yes No Yes
Cerebrovascular disease NA No NA No Yes No Yes
Chronic pulmonary disease NA No NA No Yes No Yes
Congestive heart failure NA No NA No Yes No Yes
Creatinine 0.80 1.00 1.40 0.23 3.24 0.19 3.24
Dementia NA No NA No Yes No Yes
Diabetes NA No NA No Yes No Yes
Ethnicity NA Hispanic NA Hispanic Not Hispanic Hispanic Not Hispanic
Ferritin 205.000 464.900 991.000 4.000 3646.875 3.000 3646.875
Geographic division NA Pacific NA Pacific West North Central Pacific West North Central
Heart rate 77.5 87.5 98.0 38.0 164.0 20.0 203.0
Hemiplegia or paraplegia NA No NA No Yes No Yes
Hypertension NA No NA No Yes No Yes
Lactate dehydrogenase (LDH) 230.00 308.00 422.00 65.00 1049.50 24.99 1049.50
Lymphocyte count 0.7 1.0 1.4 0.0 3.5 0.0 3.5
Metastatic cancer NA No NA No Yes No Yes
Mild liver disease NA No NA No Yes No Yes
Moderate/severe liver disease NA No NA No Yes No Yes
Neutrophil count 3.36 4.90 7.10 0.00 18.32 0.00 18.32
Oxygen saturation 94.0 96.0 98.0 58.5 100.0 26.0 100.0
Peptic ulcer disease NA No NA No Yes No Yes
Peripheral vascular disease NA No NA No Yes No Yes
Platelet count (PLT) 158 203 261 7 569 1 569
Race NA African American NA African American Caucasian African American Caucasian
Race/Ethnicity NA Non-Hispanic white NA Non-Hispanic white Non-Hispanic black Non-Hispanic white Non-Hispanic black
Renal disease NA No NA No Yes No Yes
Respiration rate 18.00000 20.00000 22.00000 5.00000 56.25904 2.00000 99.00000
Rheumatoid disease NA No NA No Yes No Yes
Sex NA Male NA Male Female Male Female
Smoking NA Never NA Never Previous Never Previous
Systolic blood pressure 115 126 139 45 225 30 266
Temperature 36.70000 37.00000 37.40000 16.00000 41.22064 16.00000 42.20000
Troponin I 0.010 0.010 0.045 0.000 0.170 0.000 0.170
White blood cell count (WBC) 4.91 6.70 9.10 0.30 21.70 0.20 21.70

Let’s make the predictions.

lrm_log_odds <- Predict(lrm_fit_all, ref.zero = TRUE)

# Get plotting data
p_log_odds <- ggplot(lrm_log_odds, sepdiscrete = "list")

# Continuous plot
log_odds_limit <- max(ceiling(abs(p_log_odds$continuous$data$yhat)))
log_odds_breaks <- seq(-log_odds_limit, log_odds_limit, 1)
p_log_odds_continuous <- p_log_odds$continuous$data %>%
  as_tibble() %>%
  mutate(varlab = get_var_labs(.predictor.)) %>%
  ggplot(aes(x = .xx., y = yhat)) +
  facet_wrap(~varlab, scales = "free_x", ncol = 4) +
  geom_line() +
  geom_ribbon(aes(ymin = lower, ymax = upper), alpha = 0.3) +
  geom_hline(yintercept = 0, linetype = "dashed", color = "grey") +
  ylab("Log odds") +
  scale_y_continuous(breaks = log_odds_breaks,
                     limits = c(-log_odds_limit, log_odds_limit)) +
  theme(axis.title.x = element_blank(), 
        strip.text = element_text(size = 7))

# Discrete plot
log_odds_limit <- max(ceiling(abs(p_log_odds$discrete$data$yhat)))
log_odds_breaks <- seq(-log_odds_limit, log_odds_limit, 1)
p_log_odds_discrete <- p_log_odds$discrete$data %>%
  as_tibble() %>%
  mutate(varlab = get_var_labs(.predictor.)) %>%
  ggplot(aes(x = yhat, y = .xx.)) +
  facet_wrap(~varlab, scales = "free_y", ncol = 4) +
  geom_point(size = .9) +
  geom_errorbarh(aes(xmin = lower , xmax = upper, height = 0)) +
  geom_vline(xintercept = 0, linetype = "dashed", color = "grey") +
  xlab("Log odds") +
  scale_x_continuous(breaks = log_odds_breaks,
                     limits = c(-log_odds_limit, log_odds_limit)) +
  theme(axis.title.y = element_blank(),
        strip.text = element_text(size = 7))

# Combine plots
grid.arrange(p_log_odds_discrete, p_log_odds_continuous,
             heights = c(5, 5))
## Warning: Removed 6 rows containing missing values (geom_errorbarh).

6.7 Predicted probabilities

The predicted log odds plot is a nice way to summarize non-linear effects, but it’s not the ultimate quantity of interest. We really care about predicted probabilities. But to compute a predicted probability we need to set all covariates in the model to certain values. Rather that choosing specific values (i.e., creating a representative patient), we will make predictions over a representative sample and average predictions across the sample.

# Make newdata
## Start with a random sample of patients
n_samples <- 1000
train_sample <- mi_list[[1]] %>% 
  filter(.imp > 0) %>%
  sample_n(size = n_samples)

We will write a general function to predict mortality as a function of (i) our fitted model, (ii) the representative sample, and (iii) the variables that we are varying.

expand_newdata <- function(data, vars_to_vary){
  varnames_to_vary <- names(vars_to_vary)
  expanded_vars_to_vary <- expand.grid(vars_to_vary)
  data %>% 
    select(-all_of(varnames_to_vary)) %>%
    crossing(expanded_vars_to_vary)
}

predict_mortality <- function(fit, newdata, vars_to_vary){
  expanded_newdata <- expand_newdata(newdata, vars_to_vary)
  fit_probs <- predict(fit, newdata = data.frame(expanded_newdata), 
                       type = "fitted", se.fit = FALSE)
  
  # Average predictions for variables to vary
  expanded_newdata %>%
    mutate(prob = fit_probs) %>%
    group_by(across(all_of(names(vars_to_vary)))) %>%
    summarise(prob = mean(prob))
}

predict_mortality_boot <- function(fit, newdata, vars_to_vary, B = 100){
  n_obs <- nrow(newdata)
  boot_probs <- vector(mode = "list", length = B)
  for (b in 1:B){
    indices <- sample(x = 1:n_obs, size = n_obs, replace = TRUE) 
    newdata_boot <- newdata[indices, ]
    boot_probs[[b]] <- predict_mortality(fit, newdata_boot, vars_to_vary)
    boot_probs[[b]][, "b"] <- b 
  }
  return(boot_probs %>% bind_rows())
}

6.7.1 Varying age

ages_to_vary  <- seq(min(train_data$age),  max(train_data$age), 1)

probs_age <- predict_mortality(lrm_fit_all, newdata = train_sample, 
                               vars_to_vary = list(age = ages_to_vary)) %>%
  rename(Age = age, `Mortality probability` = prob)
## `summarise()` ungrouping output (override with `.groups` argument)
# Table using DT package
datatable(probs_age, 
          rownames = FALSE,
          options = list(pageLength = 20)) %>%
  formatRound("Mortality probability", 4)
ggplot(probs_age, aes(x = `Age`, y = `Mortality probability`)) +
  geom_line()

6.7.2 Varying calendar time

min_index_date <- min(train_data$index_date)
calendar_times_to_vary <- c(as.Date("2020-03-01"), 
                            as.Date("2020-04-01"), 
                            as.Date("2020-05-01")) - min_index_date 
calendar_times_to_vary <- as.numeric(calendar_times_to_vary)

probs_calendar_time <- predict_mortality(
  lrm_fit_all, newdata = train_sample, 
  vars_to_vary = list(calendar_time = calendar_times_to_vary)
) %>%
  mutate(Date = min_index_date + calendar_time) %>%
  rename(`Calendar time` = calendar_time, `Mortality probability` = prob) %>%
  relocate(Date)
## `summarise()` ungrouping output (override with `.groups` argument)
kable(probs_calendar_time) %>% kable_styling()
Date Calendar time Mortality probability
2020-03-01 9 0.2276039
2020-04-01 40 0.1724317
2020-05-01 70 0.1395255

6.7.3 Varying age, sex, and calendar time

We will start by predicting point estimates only.

sex_to_vary <- unique(train_data$sex) %>%
  .[!is.na(.)]

# Predict point estimates
probs_age_sex_calendar <- predict_mortality(
  lrm_fit_all, newdata = train_sample, 
  vars_to_vary = list(age = ages_to_vary,
                      sex = sex_to_vary,
                      calendar_time = calendar_times_to_vary))
## `summarise()` regrouping output by 'age', 'sex' (override with `.groups` argument)
# Summarize in table
probs_age_sex_calendar %>%
  arrange(calendar_time, sex, age) %>%
  mutate(Date = min_index_date + calendar_time) %>%
  select(-calendar_time) %>%
  relocate(Date, sex) %>%
  rename(`Sex` = sex, `Age` = age,
         `Mortality probability` = prob) %>%
  
  ## Using DT::datatable
  datatable(rownames = FALSE,
            filter = "top",
          options = list(pageLength = 20)) %>%
  formatRound("Mortality probability", 4)

We we will also compute 95% confidence intervals by bootstrapping.

bootprobs_age_sex_calendar <- predict_mortality_boot(
  lrm_fit_all, newdata = train_sample, 
  vars_to_vary = list(age = ages_to_vary,
                      sex = sex_to_vary,
                      calendar_time = calendar_times_to_vary),
  B = n_boot_probs)

Let’s plot the results.

bootprobs_age_sex_calendar %>%
  # Summarize bootstrap results
  group_by(age, sex, calendar_time) %>%
  summarise(
    prob_mean = mean(prob),
    prob_lower = quantile(prob, .025),
    prob_upper = quantile(prob, .975)
  ) %>%
  mutate(date = min_index_date + calendar_time) %>%

  # Plot
  ggplot(aes(x = age, y = prob_mean, color = sex, fill = sex)) +
    geom_line() +
    facet_wrap(~date) +
    geom_line(aes(x = age, y = prob_lower, color = sex),  linetype = "dotted") +
    geom_line(aes(x = age, y = prob_upper, color = sex),  linetype = "dotted") +
    xlab("Age") +
    ylab("Mortality probability") +
    scale_color_discrete("") +
    scale_fill_discrete("") +
    theme(legend.position = "bottom")
## `summarise()` regrouping output by 'age', 'sex' (override with `.groups` argument)

7 Model validation

7.1 Optimism-adjusted model performance

We will start by using bootstrapping to estimate model performance and check for whether our in-sample fits are too optimistic. Specifically, we will use the following algorithm implemented in the rms package:

  1. Calculate performance from the model fit on the original training sample, denoted as \(\theta_{orig}\).
  2. For \(b = 1,\ldots,B\):
    1. Bootstrap (with replacement) from the original training sample.
    2. Fit the model to the bootstrapped data and estimate model performance, denoted as \(\theta_{train}\).
    3. Use the bootstrap fit to measure model performance on the original training sample, denoted as \(\theta_{test}\).
  3. Calculate “optimism” as \(O = B^{-1} \sum_{b = 1}^B \theta_{b, train} - \theta_{b, test}\).
  4. Calculate optimism-adjusted performance as \(\theta = \theta_{orig} - O\).

A shrinkage factor can also be estimated within each bootstrap sample to gauge the extent of overfitting. This is done by fitting \(g(Y) = \gamma_0 + \gamma_1 X\hat{\beta}\) where \(X\) and \(Y\) are the covariates and outcome, respectively, in the test sample (i.e., in step 2c) and \(\hat{\beta}\) is estimating in the training sample (i.e., step 2b). If there is no overfitting, then \(\gamma_0 = 0\) and \(\gamma_1 = 1\); conversely, if there is overfitting, then \(\gamma_1 < 1\) and \(\gamma_0 \neq 1\) to compensate.

lrm_val_age <- validate(lrm_fit_age, B = n_boot_val)
lrm_val_d <- validate(lrm_fit_d, B = n_boot_val)
lrm_val_dc <- validate(lrm_fit_dc, B = n_boot_val)
lrm_val_all <- validate(lrm_fit_all, B = n_boot_val) 
bind_cindex <- function(object){
  n_rows <- nrow(object)
  c_index <- (object["Dxy", 1:3] + 1)/2
  c_index[4] <- c_index[2] - c_index[3]
  c_index[5] <- c_index[1] - c_index[4]
  c_index[6] <- object[1, 6]
  
  return(rbind(object, c_index))
}

make_validation_tbl <- function(object){
  object %>% 
    bind_cindex() %>%
    set_colnames(c("(1) Original", "(2) Bootstrap training", 
                   "(3) Bootstrap test", "Optimism: (2) - (3)", 
                   "Original (corrected): (1) - (4)", "N")) %>%
    kable() %>%
    kable_styling()
}

make_validation_tbl(lrm_val_age)
  1. Original
  1. Bootstrap training
  1. Bootstrap test
Optimism: (2) - (3) Original (corrected): (1) - (4) N
Dxy 0.5492928 0.5506700 0.5492928 0.0013772 0.5479156 40
R2 0.2121268 0.2135547 0.2121268 0.0014279 0.2106989 40
Intercept 0.0000000 0.0000000 -0.0065005 0.0065005 -0.0065005 40
Slope 1.0000000 1.0000000 0.9947905 0.0052095 0.9947905 40
Emax 0.0000000 0.0000000 0.0023089 0.0023089 0.0023089 40
D 0.1318066 0.1328923 0.1318066 0.0010856 0.1307210 40
U -0.0001463 -0.0001463 -0.0000331 -0.0001133 -0.0000331 40
Q 0.1319530 0.1330386 0.1318397 0.0011989 0.1307541 40
B 0.1159272 0.1159854 0.1159503 0.0000350 0.1158922 40
g 1.3965451 1.4025119 1.3965451 0.0059668 1.3905783 40
gp 0.1457790 0.1464509 0.1457790 0.0006719 0.1451070 40
c_index 0.7746464 0.7753350 0.7746464 0.0006886 0.7739578 40 
make_validation_tbl(lrm_val_all)
  1. Original
  1. Bootstrap training
  1. Bootstrap test
Optimism: (2) - (3) Original (corrected): (1) - (4) N
Dxy 0.7819128 0.7840229 0.7778674 0.0061555 0.7757573 40
R2 0.4571773 0.4614948 0.4528691 0.0086258 0.4485516 40
Intercept 0.0000000 0.0000000 -0.0201162 0.0201162 -0.0201162 40
Slope 1.0000000 1.0000000 0.9766515 0.0233485 0.9766515 40
Emax 0.0000000 0.0000000 0.0086833 0.0086833 0.0086833 40
D 0.3095606 0.3125432 0.3061459 0.0063972 0.3031633 40
U -0.0001463 -0.0001463 0.0000799 -0.0002263 0.0000799 40
Q 0.3097069 0.3126895 0.3060660 0.0066235 0.3030834 40
B 0.0869979 0.0863768 0.0877119 -0.0013351 0.0883330 40
g 2.4021654 2.4334329 2.3763771 0.0570558 2.3451096 40
gp 0.2053625 0.2057372 0.2044055 0.0013317 0.2040308 40
c_index 0.8909564 0.8920115 0.8889337 0.0030778 0.8878786 40 
summarize_performance <- function(objects){
  metrics <- c("Dxy", "R2", "B")
  map_df(objects, function (x) x[metrics, "index.orig"],
                .id = "Model") %>%
    mutate(`C-index` = (Dxy + 1)/2) %>%
    rename(`Brier score` = B, `R-Squared` = R2) %>%
    select(-Dxy) %>%
    kable(digits = 3) %>%
    kable_styling()
}

list(lrm_val_age, lrm_val_d, lrm_val_dc, lrm_val_all) %>%
  set_names(lrm_names) %>%
  summarize_performance()
Model R-Squared Brier score C-index
Age only 0.212 0.116 0.775
All demographics 0.229 0.114 0.785
Demographics and comorbidities 0.256 0.112 0.803
All variables 0.457 0.087 0.891

7.2 Calibration curves

lrm_cal_age <- calibrate(lrm_fit_age, B = n_boot_val)
lrm_cal_d <- calibrate(lrm_fit_d, B = n_boot_val)
lrm_cal_dc <- calibrate(lrm_fit_dc, B = n_boot_val)
lrm_cal_all <- calibrate(lrm_fit_all, B = n_boot_val)
lrm_cal_list <- list(lrm_cal_age, lrm_cal_d, lrm_cal_dc, lrm_cal_all)
names(lrm_cal_list) <- lrm_names

plot_calibration <- function(object){
  # Make tibble
  cal_df <- map2(object, names(object), function(x, y){
    x[, ] %>%
      as_tibble() %>%
      mutate(model = y)
  }) %>% 
    bind_rows() %>%
    mutate(model = factor(model, levels = lrm_names))
  
  
  # Plot
  breaks <- seq(0, 1, .2)

  ggplot() + 
    geom_line(data = cal_df, mapping = aes(x = predy, y = calibrated.orig, 
                                           color = "Apparent")) + 
    geom_line(data = cal_df, mapping = aes(x = predy, y = calibrated.corrected,
                                           color = "Bias-corrected")) +
    geom_abline(intercept = 0, slope = 1, linetype = "dashed", color = "grey") +
    facet_wrap(~model) +
    scale_x_continuous(breaks = breaks, limits = c(0, 1)) +
    scale_y_continuous(breaks = breaks, limits = c(0, 1)) +
    xlab("Predicted probability") +
    ylab("Actual probability") +
    scale_colour_manual(name = "",
                        values = c("Apparent" = "black", 
                                   "Bias-corrected" = "red")) +
    theme(legend.position = "bottom") 
}

plot_calibration(lrm_cal_list)

7.3 Performance on test set